In some books on category theory (for example, in J.Adámek, H.Herrlich, E.Strecker "Abstract and concrete categories...") the authors use the idea of "big sets" ("conglomerates" or "collections") which can contain classes (as far as I understand, in the Goedel-Bernays sense) as elements, and they formulate the "generalized axiom of choice", where it is stated that the choice function exists (not only for families or classes of sets, but also) for families of classes (indexed by elements of those "big sets"). This approach allows to prove, in particular, the existence of a skeleton in each category, and some other useful things.

I wonder if there are any texts with the justification of this trick? The references I found (in particular, those mentioned in Wikipedia) give justification only for usual axiom of choice (for families of sets or for classes of sets, but not for "conglomerates of classes"). So actually I can't understand whether, for example, the existence of a skeleton, is true for all categories (in some interpretation of set theory) or for some special ones... Similarly the other corollaries of this "global axiom of choice" look doubtful for me. I would be grateful if anybody could clarify this.

UPDATE 21.09.2012

From the comments I see that there is a risk of misunderstanding, so I want to explain that by justification I mean an accurate (rigorous) definition of the new tool together with the analysis of whether it is compatible with the other tools of the theory.

As an illustration, in the case of the usual axiom of choice (I mean its "weak form", in terms of Wikipedia), there are many textbooks (I can recommend E.Mendelson "Introduction to mathematical logic" or J.Kelly "General topology", the appendix), where the fundamental objects of the theory (in this case, the classes) are accurately introduced (here, axiomatically) and the necessary constructions (like functions) are rigorously defined in the theory. This makes possible to give rigorous formulation to the axiom of choice (again, to its "weak form") inside the theory, and moreover, this presentation of a new axiom is followed by a thorough investigation of whether it contradicts to the previous axioms of the theory. Only after receiving the answer that no contradictions can appear (in fact, a more strong thing is true: the new axiom is independent from the others, that was the result by P.Cohen) mathematicians can use this new axiom without worrying that something is wrong here.

So my question is whether there is something similar for the "strong form of the axiom of choice"? Is it possible that nothing lies behind these words? On the contrary, if there is a justification, where can I read about it?

UPDATE 21.04.2013

Dear colleagues, from what I learn on this subject in the textbooks which I found, in Wikipedia and here in MO, I deduce that what people call "axiom of global choice" is just the usual axiom of choice (as it is presented in Kelly's book) applied to some special classes of sets arising in consideration of what is called the Grothendieck Universe. It's a puzzle for me

1) why people call this special case "a stronger form of the axiom of choice", and

2) why they don't want to give references, where this construction is accurately introduced.

With the aim to accelerate the clarification of this question, I now nominated for deletion the article in Wikipedia devoted to his topic: http://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Axiom_of_global_choice. As I wrote there, I don't exclude that the partisans of the idea will rewrite the article in Wikipedia for endowing "global choice" with some mathematical sense, but you should agree that in its present form this article and the other mentionings of "global choice" available for external observers, look indecently vague. I invite all comers to share their opinion here or at the Wikipedia page.

One way of putting it is this: the axiom of choice for sets is equivalent to the statement that every small category has a skeleton. Since the definition of category is first order, you could consider the axiom: Every model of the category axioms has a skeleton. This is then prior to any choice of ambient set theory.
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David RobertsSep 20 '12 at 7:27

5

What exactly do you mean by "justification" of an axiom? I'd use that phrase to mean the pre-axiomatic intuitive ideas that lead me to regard the statement as a reasonable axiom to include in my theory. For this purpose, it seems to me that whatever intuition leads you to accept the axiom of choice for sets would probably do the same for classes, conglomerates, and whatever higher entities you include in your theory.
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Andreas BlassSep 20 '12 at 13:25

2

I don't know that much about set theory, but I guess to get a workable theory of conglomerates (in ZF+something) we need some large cardinal axiom and then the axiom of choice is simply the usual axiom of choice, applied to very large sets then. And if the most intriguing part is "how can a class (which is not a set) be an element of someting else?", then this has little to do with the axiom of choice.
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Michael GreineckerSep 21 '12 at 10:35

6

Sergei: The main reason why you're not getting an answer is not necessarily the meaning of "justification" but the meaning of "conglomerate." None of the standard set theories (ZF, NBG, MK) admit those. Some more esoteric theories do (e.g. Ackermann) but since these can be wildly different from each other, you really need to say which one you're using before any kind of serious analysis can be done. If you don't know which one you are using then you can ask a separate question to figure that out first.
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François G. Dorais♦Sep 21 '12 at 12:21

3 Answers
3

Here at least is the usual justification for moving from AC for sets to what is normally called the global axiom of choice, which asserts that there is a class well-ordering of the (first-order) universe.

Theorem.

The global axiom of choice, when added to the ZFC or
GB+AC axioms of set theory, leads to no new theorems about sets.
That is, the first-order assertions about sets that are provable
in GBC are precisely the same as the theorems of ZFC.

Furthermore, every model of ZFC can be extended (by forcing) to a
model of GBC, in which the global axiom of choice is true, while
adding no new sets (only classes).

In particular, the global axiom of choice is safe in the
sense that it will not cause inconsistency, unless the underlying
system without the global axiom of choice was already inconsistent.

Proof. Suppose that $M$ is any model of ZFC. Consider the class
partial order $\mathbb{P}$ consisting of all well-orderings in $M$
of any set in $M$, ordered by end-extension. As a forcing notion,
this partial order is $\kappa$-closed for every $\kappa$ in $M$,
since the union of a chain of (end-extending) well-orderings is
still a well-order. If $G\subset\mathbb{P}$ is $M$-generic for
this partial order, then $G$ is, in effect, a well-ordering of all
the sets in $M$. Furthermore, one can prove by the usual forcing
technology that the structure $\langle M,{\in},G\rangle$ satisfies
$\text{ZFC}(G)$, that is, where the predicate $G$ is allowed to
appear in the replacement and other axiom schemes.

Essentially, what we've done is add a global well-ordering of the
universe generically. And since the forcing was closed, no new
sets were added, and so $M[G]$ has the same first-order part as
$M$.

It follows now that GBC is conservative over ZFC for first-order
assertions, since any first-order statement $\sigma$ that is true
in all GBC models will be true in $M[G]$ and therefore also in
$M$, and so $\sigma$ is true in all ZFC models as well. QED

I am afraid, this is too technical for me. Are you saying that it is possible to extend the Gödel-Bernays theory in such a way that the axiom of choice becomes more powerful, it can be applied to classes?
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Sergei AkbarovApr 13 '13 at 16:39

2

I might add that this is completely standard material when considering Goedel-Bernays set theory, and so if you do find this "completely unfamiliar", then it may be appropriate for you to adopt a less strident tone.
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Joel David HamkinsApr 21 '13 at 11:30

2

In particular, the Wikipedia page on global choice en.wikipedia.org/wiki/Axiom_of_global_choice seems to me currently to be completely fine, presenting the standard and familiar facts about it as they are usually understood, including the conservation result that I prove above. Please do not delete that Wikipedia page, and please remove your request for deletion.
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Joel David HamkinsApr 21 '13 at 11:40

1

The axiom of choice for classes (i.e. global choice) is known not to be equivalent to the axiom of choice for sets, as one can build a model of Goedel-Bernays set theory that does not satisfy the axiom of choice for classes but does satisfy AC (this is done in a few questions here on MO). For this reason, I would regard it as a mistake to refer to both axioms as the "axiom of choice". Indeed, such a terminology would likely cause students to become very confused about the difference between them.
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Joel David HamkinsApr 24 '13 at 19:17

1

Well, the forcing arguments show that global choice (choice for classes) is strictly stronger than the axiom of choice (choice for sets). I don't find that to be banal at all; rather, I find it to be a very enlightening piece of mathematics, explaining something fundamental about the set/class distinction. I would suggest that you take a look at it...
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Joel David HamkinsApr 25 '13 at 20:23

I think that you should place yourself in ZFC+ existence of strongly innaccessible cardinals.

Then the existence of a strongly inaccessible cardinal provides you a universe as in Borceux's Handbook of Categorical algebra.

Then, what you call sets are elements of the universe, and what you call classes are the subsets of the universe, but they are still sets in the set theoretic sense, so you can apply choice.

EDIT: clarification

The problem of category theory is that we want to have the category Set of all sets to actually be a category.

Since there is no set which contains all sets, we can't ask a category to have a set of objects, or Set will no more be a category. That's why in the first place we define the collection of object to be potentially wider than a set: we ask it to be a class.

The point is that you can avoid the difficulty differently, by limiting yourself to a rich enough set of sets, which should contains "everthing that you can be interested in".

Existence of Grothendieck universes turns out to be equivalent to the existence of strongly inaccessible cardinals (here we are in ZFC), and this existence axiom has been studied in set theory (I'm not a specialist of that at all).

So you place yourself in ZFC + existence of strongly inaccessible cardinals, and you take a universe $U$. Call the elements of U the "U-sets" and the subsets of U the "U-classes".

Dimitri, I did not undertand you. Is this written anywhere? If not, can you contact me to explain what you write? (Or explain this here.)
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Sergei AkbarovApr 8 '13 at 5:18

I hope my edit made the ideas I tried to express clearer.
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Dimitri ZaganidisApr 13 '13 at 15:39

Dimitri, is it possible to translate this into the language of the Gödel-Bernays set theory?
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Sergei AkbarovApr 13 '13 at 16:15

As far as I understand, $U$ must be a set here, is it? If yes, then I don'see any profits. If the "axiom of global choice" is just the usual axiom of choice applied to families of $U$-classes (which are families of sets), then why do people introduce the very term "axiom of global choice", and claim that "the axiom of global choice is a stronger variant of the axiom of choice which applies to proper classes as well as sets" (en.wikipedia.org/wiki/Axiom_of_global_choice). This is evidently not true in the situation you describe.
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Sergei AkbarovApr 20 '13 at 8:05

Dear Sergei, you might be interested in first reading Bourbaki's Théorie des ensembles (at least chapters I--III) and then have a look at section 0 and the appendix of SGA 4.I. This gives a slightly different approach using Hilbert's almighty symbol $\tau$.